Bifurcations of Homoclinic Orbit Connecting Two Nonleading Eigendirections

Abstract

Bifurcations of homoclinic orbit connecting the strong stable and strong unstable directions are investigated for four-dimensional system. The existence, numbers, co-existence and incoexistence of 1-homoclinic orbit, 2n-homoclinic orbit, 1-periodic orbit and 2n-periodic orbit are obtained, and the bifurcation surfaces (including codimension-1 homoclinic bifurcation surfaces, double periodic orbit bifurcation surfaces, homoclinic-doubling bifurcation surfaces, period-doubling bifurcation surfaces and codimension-2 triple periodic orbit bifurcation surface, and homoclinic and double periodic orbit bifurcation surface) and the existence regions are also located.

Bifurcation Analysis of Homoclinic Flips at Principal Eigenvalues Resonance

“…From the above hypotheses, the normal form theory provides a system as follows after four successive to transformations in U (see [10,11,18])…”

“…Homburg and Oldeman studied two kinds of resonant homoclinic flips in [8,9] with unfolding techniques and numerical methods respectively. Zhang in [10,11] continued to research on these problems and gave some theoretical proofs of the existence of -periodic orbit and -homoclinic orbit and also their existence regions via the method initially established in [18]. Besides these the flip heterodimensional cycles have also attracted attentions nowadays, see [16].…”

“…After that many flips cases attract researcher's interests, including resonant eigenvalues case in [2], orbit flips in [3,4], inclination flips in [5][6][7], and also resonant homoclinic flips in [8][9][10][11]. In these cases homoclinic-doubling bifurcation has been expensively studied, which is a codimension-two transition from an n-homoclinic to a 2n-homoclinic orbit.…”

Resonant Homoclinic Bifurcations with Orbit Flips and Inclination Flips

“…Firstly system (1.1) can be transformed into the following form in some neighborhood of the origin O due to the theory of invariance manifolds, (refer to [14,15])…”

“…In this paper, we develop a study of resonant homoclinic bifurcation with one orbit flip and two inclination flips, where the resonance takes place in the tangent direction of the homoclinic orbit. This is a codimension-4 problem, by using the local moving frame method established in [11,14,15], we get the existence of a double 1-periodic orbit, some 1-periodic orbits and 1-homoclinic orbits, and the coexistence conditions of 1-periodic orbits and 1-homoclinic orbits.…”